the category of T0 Alexandroff spaces is equivalent to the category of posets


Let 𝒜𝒯 be the category of all T0, Alexandroff spaces and continuous maps between them. Furthermore let 𝒫𝒪𝒮𝒯 be the category of all posets and order preserving maps.

Theorem. The categories 𝒜𝒯 and 𝒫𝒪𝒮𝒯 are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

Proof. Consider two functors:

T:𝒜𝒯𝒫𝒪𝒮𝒯;
S:𝒫𝒪𝒮𝒯𝒜𝒯,

such that T(X,τ)=(X,), where is an induced partial orderMathworldPlanetmath on an Alexandroff space and T(f)=f for continuous map. Analogously, let S(X,)=(X,τ), where τ is an induced Alexandroff topologyMathworldPlanetmath on a poset and S(f)=f for order preserving maps. One can easily show that T and S are well defined. Furthermore, it is easy to verify that equalities TS=1𝒫𝒪𝒮𝒯 and ST=1𝒜𝒯 hold, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Remark. Of course every finite topological space is Alexandroff, thus we have very nice ,,interpretationMathworldPlanetmathPlanetmath” of finite T0 spaces - finite posets (since functors T and S do not change set-theoretic properties of underlying sets such as finitness).

Title the category of T0 Alexandroff spaces is equivalent to the category of posets
Canonical name TheCategoryOfT0AlexandroffSpacesIsEquivalentToTheCategoryOfPosets
Date of creation 2013-03-22 18:46:04
Last modified on 2013-03-22 18:46:04
Owner joking (16130)
Last modified by joking (16130)
Numerical id 8
Author joking (16130)
Entry type Theorem
Classification msc 54A05